Calculus Examples

$\ln (\sqrt[3]{3 r^{2} s})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{3 r^{2} s}$ as ${(3 r^{2} s)}^{\frac{1}{3}}$ .

$\ln ({(3 r^{2} s)}^{\frac{1}{3}})$

Step 2

Expand $\ln ({(3 r^{2} s)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \ln (3 r^{2} s)$

Step 3

Rewrite $\ln (3 r^{2} s)$ as $\ln (3 r^{2}) + \ln (s)$ .

$\frac{1}{3} (\ln (3 r^{2}) + \ln (s))$

Step 4

Rewrite $\ln (3 r^{2})$ as $\ln (3) + \ln (r^{2})$ .

$\frac{1}{3} (\ln (3) + \ln (r^{2}) + \ln (s))$

Step 5

Expand $\ln (r^{2})$ by moving $2$ outside the logarithm.

$\frac{1}{3} (\ln (3) + 2 \ln (r) + \ln (s))$

Step 6

Apply the distributive property.

$\frac{1}{3} \ln (3) + \frac{1}{3} (2 \ln (r)) + \frac{1}{3} \ln (s)$

Step 7

Simplify.

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Step 7.1

Combine $\frac{1}{3}$ and $\ln (3)$ .

$\frac{\ln (3)}{3} + \frac{1}{3} (2 \ln (r)) + \frac{1}{3} \ln (s)$

Step 7.2

Multiply $\frac{1}{3} (2 \ln (r))$ .

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Step 7.2.1

Combine $2$ and $\frac{1}{3}$ .

$\frac{\ln (3)}{3} + \frac{2}{3} \ln (r) + \frac{1}{3} \ln (s)$

Step 7.2.2

Combine $\frac{2}{3}$ and $\ln (r)$ .

$\frac{\ln (3)}{3} + \frac{2 \ln (r)}{3} + \frac{1}{3} \ln (s)$

Step 7.3

Combine $\frac{1}{3}$ and $\ln (s)$ .

$\frac{\ln (3)}{3} + \frac{2 \ln (r)}{3} + \frac{\ln (s)}{3}$